A prior art pressure-measuring arrangement of the above kind is shown in FIGS. 17 to 19. A pressure sensor structure, generally designated by the reference numeral 1 and shown in a top view in FIG. 17 and in a schematic cross-sectional view in FIG. 19, comprises a sensor body 2 which defines a reference-pressure space 3 that is covered by a diaphragm 4. The diaphragm 4 comprises an inner, circular electrode 5, which forms with a (respective) counter-electrode (not visible in FIG. 17) on the sensor body 2 a pressure-dependent measuring capacitance C.sub.s, and an outer, substantially annular electrode 6, which forms with the counter-electrode an essentially pressure-independent reference capacitance C.sub.r.
An evaluating circuit for the prior art pressure-measuring arrangement is illustrated in FIG. 18 and includes a DC voltage source U.sub.G. The reference capacitance C.sub.r is connectable via a first switching element S1 either to the DC voltage source UG or to the inverting input of an operational amplifier OPV, whose noninverting input is grounded. A capacitor C.sub.K is connected between the inverting input and the output of the operational amplifier OPV.
One electrode of the measuring capacitance C.sub.s is connectable either to the inverting input or to the output of the operational amplifier OPV by means of a second switching element S2, while the other electrode of the measuring capacitance C.sub.s is grounded.
A summing point SP is supplied with the voltage of the DC voltage source U.sub.G and with the output voltage, negative in sign, of the operational amplifier OPV. It is evident to those skilled in the art that the output voltage of the operational amplifier is proportional to the reference capacitance C.sub.r and inversely proportional to the measuring capacitance C.sub.s. Since this output voltage is applied to the summing point SP with negative sign, the prior art circuit has the following transfer function F: EQU F=(C.sub.s -C.sub.r)/C.sub.s
The following derivation will show that the curvature of the diaphragm 4 supporting the electrodes 5, 6 and shown very enlarged in FIG. 19 results in a nonlinearity of the output signal which is dependent on the pressure to be measured.
For the deflection w(r) of the diaphragm, the following relation holds, assuming that the thickness h of the diaphragm 4 is much smaller than its diameter and greater than the deflection w: EQU w(r)=p(R.sup.2 -r.sup.2).sup.2 /(64D) (1)
where r is the radius under consideration, R is the radius of the diaphragm where it is fixed to the sensor body 2--in the following this radius will be designated "outer radius"--, p is the pressure, and D is the flexural strength. The latter is given by EQU D=Eh.sup.3 /[12(1-67.sup.2)] (2)
where E is the modulus of elasticity, h is the thickness of the diaphragm (see FIG. 19), and .delta. is Poisson's ratio.
For the pressure-dependent sensor capacitance C.sub.s (p), the following integral holds: ##EQU1## where r*=r/R is the normalized radius--therefore, at the sensor body 2 holds: r*=1--, and .epsilon..sub.0 is the permitivity of vacuum. Solving the integral yields the following pressure dependence of the sensor capacitance C.sub.s (p)--being Equation (4)--: ##EQU2##
Equation (4) includes the basic capacitance C.sub.0 and the support pressure p.sub.0 as newly introduced constants. For these quantities, the following relations hold: EQU C.sub.s (0)=r*.sup.2 C.sub.0 ( 5) EQU C.sub.0 =.epsilon..sub.0 .pi.R.sup.2 /d (6) EQU p.sub.0 =16dEh.sup.3 /[3R.sup.4 (1-.delta..sup.2)] (7)
It is apparent from the transfer function F of the evaluating circuit of FIG. 18 and from the pressure dependence of the pressure capacitance C.sub.s given in Equation (4) that the prior art pressure-measuring arrangement exhibits a nonlinear relationship between output voltage and pressure.
Since, to a first approximation, the characteristic of the sensor capacitance is hyperbolic, a certain linearization can be produced by forming the reciprocal, which is done in the prior art circuit of FIG. 18 by inserting the sensor capacitance Cs into the feedback path of the evaluating circuit. Such a prior art circuit is about four to five times more linear than pressure-measuring arrangements in which the measuring capacitance and the reference capacitance are located at the input and in the feedback path of an evaluating circuit respectively.
Since, however, the characteristic of the measuring capacitance or sensor capacitance C.sub.s is not exactly hyperbolic, it is not possible to generate a zero of the error function with a pressure-measuring arrangement as shown in FIGS. 17 to 19.
The publication U. Schoneberg et al., "A CMOS-Readout Amplifier For Instrumentation Applications", . . . , Ed. Frontieres 1990, pages 208 to 217, shows a pressure sensor arrangement with an evaluating circuit having a transfer function which is proportional to the pressure-dependent measuring capacitance less a pressure-independent reference capacitance. The further capacitance given in the denominator of the transfer function is a constant quantity. This circuit serves to measure the capacitance value of capacitive sensors, and thus also of capacitive pressure sensors. This pressure-sensor arrangement with a capacitive pressure sensor and the evaluating circuit is designed for a single pressure-dependent capacitance, designated there by the reference characters CSEN1, CSEN2. All other capacitances of the prior art evaluating circuit are constant, pressure-independent quantities.
As was explained above, this prior art pressure-measuring arrangement has an output voltage which is nonlinear because of the nonlinear relation between pressure and sensor capacitance.
Based on this prior art, the invention has for its object to provide a pressure-measuring arrangement which exhibits increased linearity.